Let's assume we are working with discrete data and we are interested in computing DPMO and Yield. The following is given:

Total defects = 100

Opportunity for defect per unit = 1

Number of units = 1000

There are two approches to compute DPMO.

Approach 1

DPMO = (Total defects)*1000000/(Total opportunities for defects)

DPMO  = 100*1000000/1000 = 100,000

Yield = 90%

Approach 2

There is another approach using the Poissong approximation

DPMO = (1-exp(-DPU))*1000000

DPU = 0.1

DPMO = 95,162

Yield = 90.48%

What is the difference between these two approaches? Which is the right way to do it? 

Approach 2 is more accurate allright, but does it work only when OFE is 1?

Dear Ram,

The formula as it stands in Approach 2 is derived for OFE = 1. We can derive similar formulae when OFE is not equal to 1. Let's assume OFE = 1 for now and answer the question with respect to Approach 1 and Approach 2.

Any other thoughts from the rest of the members?

SJ.

Hi Sj sir & Mr Ram,

Approach 1 is suitable, till you are sure that process is robust and stable for poisson distribution.

Also, while both approaches are correct, I would prefer the second one for accurate confirmation first. In the case of DPMO, the views on opportunities, it requires correct QA determination of the opportunities, you could make a mistake sometimes. In that case, the second method put you in the right sigma calculations and then do the reverse way of calculating DPMO to be sure what you missed in your checkpoints.

If the OFE is in a million opportunities, the level of error/defects is already below 3.4 (6 sigma parameter), which meets & surpasses the expectation. Would appreciate other thoughts.

Thnx & Rgds,

Manian

Hi again,

           The last para in my pervious reply refers 2 OFE as "1" in a million opportunities (as mentioned by Mr Ram which is below 3.4 defects) which escaped mention therein. Sorry abt that.

Rgds,

Manian.

Poisson distribution is valid if

• data are counts of discrete events
  
• defects occur independently
  
• there is equal opportunity for occurrence of defects
  
• defects occur rarely
  
• area of opportunity is constant from sample to sample
  

In most cases, these assumptions are usually valid for defects, so it is safe to use Poisson approximation when we are working with defects.

When we use the Poisson formula for computing DPMO and yield, we are using the shape of the Poisson distribution to predict what is the probability of zero defects.

Probability of 0 defects = exp(-DPU).

As an illustration, if the defects for the 10 units are as shown below:

0 0 0 0 1 1 2 0 0 0

Then, the probability of zero defects = 7/10 = 70% (Yield). It only counts those units that have zero defects. Note: that you can have units with > 1 defects for Poisson!

However, if we use approach 1 and we know there are 4 defects in 10 units, then we represent the defects as follows:

0 0 0 0 1 1 1 1 0 0

Thus, the probability of zero defects = 6/10 = 60% (Yield).

The Poisson distribution will always have a higher yield because you could potentially have units with more than 1 defect. Thus, for yield calculations, the Poisson formula gives us an estimate of units that have "really" zero defects.

On the other hand, when we calculate DPMO, the Poisson formula gives us 3 defects as 4 defects are "packed" into 3 units. So, DPMO = 300,000 (based on 1 - yield).

Approach 1 counts each defect separately, so DPMO = 400,000. From a Six Sigma point of view, we want to focus on and eliminate defects. We don't want to ignore defects if they all fall on one unit. Hence, approach 1 is preferable for calculation of DPMO as it gives us higher numbers by considering each defect independently.

My conclusion is:

Use the DPMO formula (approach 1) for calculation of the DPMO even if the data follows Poisson distribution as this approach counts all defects and not just units that have one or more defects.

Use the Poisson approximation (approach 2) for the calculation of Yield if the data follows Poisson distribution as this approach correctly counts the units that have zero defects by considering units that have one or more defects.

What are your thoughts on the above?

Let's now look at the case when OFE (Opportunities for Error) is not equal to 1. Let's look at a similar example that we looked at earlier. You can think of OFE in terms of steps in a process. Assume there are 10 steps in a process connected sequentially and in each step there is a chance of getting a defect. For the overall 10 step process, you have an opportunity to get 10 defects per unit.

Total defects = 100

Opportunity for defect per unit = 10

Number of units = 1000

There are two approches to compute DPMO.

Approach 1

DPMO = (Total defects)*1000000/(Total opportunities for defects)

DPMO  = 100*1000000/(1000*10) = 10,000

Yield = 99%

Sigma Level correponding to DPMO of 10,000 = 2.32 (from long-term tables)

Approach 2

Using the Poisson approximation

DPMO = (1-exp(-DPU))*1000000

DPU = 0.1

DPMO = 95,162

Yield = 90.48%

Sigma Level for DPMO of 95,162 = 1.31 (from long-term tables)

What is the right DPMO & Sigma level for the process?

Hi SJ sir,

Per your first thought & reply on the earlier opinion expressed, what I meant is, it is easy to calculate DPU and in turn First-pass or rolled throughput yield easy. We should use the sigma table to find out Sigma and in turn DPMO. This helps us like Math situation to cross verify whether the DPMO we calculated looking at defects and then at different ways/components etc...sometimes we could make wroking assumptions or others could differ our view point. DPU and Yield cannot be disputed as it is.

scrap/defect based calculations are pretty much straight forward. If you take the case of Motorola, they went to the level of taking Defect or CTQ as Pager working or not working instead of finding out how many components inside and how many ways you can see the defects etc.....It is purely a target we want to set to reach Six Sigma level of quality (3.4 DPMO) which is close to zero defect.

With due respect 2 all the valuable insights from your end in this regard, let's not mix the validity of process sigma with the type of distributions when we are calculating DPMO. Process sigma is a way of connecting yield, DPMO which in turn is an indirect measure of variability and opportunities. Irrespective of distributions control charts have always with limits tried to be fixed as 3 sigma limits + and - for the nature of measurements we do. So we can say it won't, rather may not alter the DPMO calculations.

Per your second question with the example, if we Use the sigma table it is simple. Let's not get into long-term sigma calculations in the define phase of DPMO. It may be thought to be should be used with SPC charts and capability calculations, if we want to find out Cp,Cpk as well as Pp, Ppk, if I'm not wrong.

It can be thought here, that the opportunity for defects per unit could be wrong giving two different results for us. This one should be redefined based on the product and CTQ properly. In that case both will match. This is where we should play safe with the second approach which gives us a yield of 90.48% and defect ratio also matches with that 100/1000. Now we should find out based on the product CTQ where we missed the numbers for opportunities as otherwise we will be making a grave error of sigma values.

Definitely, 99% is too high here compared to what it should be as 90.48. One can be 100% sure that process yield is only 90.48% and that can't be altered. If we use 1 instead of 10, definitely it gets blown ten times to give us a higher yield and lower DPMO which will not be the actual observed case. We need to tally with observed values of yield only. That is why it may be suggested to always use approach 2.

One more thing to note is we are looking for opportunities based on the available data under DMAIC which means, where there is an occurrence of problem or issue earlier, only those are opportunities to begin with.

In the given case, yield of 90.48% is right and the sigma level is 2.81. Sigma level calculator connects DPMO to Sigma. What we may ideally do is use the process yield and use the sigma table to find DPMO and calculate the exact sigma then with the calculator, if we want to report precise.

In this case we may think that is the way we look into. If we want to look into all the components and make it still robust (which is what Motorla must have done internally), then we should use FMEA for each component and work for DFSS/DMADV.

Under DMAIC we need to confine only to the problem area and associated process only. Strictly we are talking about variability on established processes which were earlier under some sort of control or doing SPC.

Thnx & rgds.

Dear Manian,

Thanks for your detailed response. Here are my thoughts...

The yield formula given by Approach 1 is sort of like the First Time Yield (FTY) and the yield formula given by Approach 2 is the Rolled Throughput Yield (RTY).

FTY = 99% and RTY = 90.48%.

Since we have 10 steps in the process, we can verify that roughly

FTY^10 = 0.99^10 = 90.44% which is very close to the value from Poisson's approximation.

Another way to look at it is:

We can also think of RTY^(1/n) = Yn (normalized yield)

Where, n is the number of steps in the process.

I agree with your viewpoint that people manipulate the number of steps in the process to falsely report high sigma levels. Hence, some people like to use OFE = 1 and not worry about the number of steps. On the positive side, this number cannot be manipulated but on the flip side, we will not be able to compare yields or sigma levels of different processes that have different number of steps.

In conclusion,

if we like to compare process performance across different processes (which is what the DPMO and Sigma level were originally supposed to do), then we are stuck with working with opportunities per unit. The problem here is that people start manipulating the opportunities per unit, so we can't rely on the Sigma levels anyway.

So, it may be better to work with OFE = 1 and just report overall yield (RTY) and sigma levels. But make sure that you don't use these numbers to compare processes with different number of steps - say Sigma level of a Coffee making with TV making. You can only use it to compare before and after performance for the same process.

Other thoughts...